Coordinate systems and distributional embeddings in Bourgain-Rosenthal-Schechtman spaces: a framework for operator reduction
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866918242910470144 |
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| author | Konstantos, Konstantinos Motakis, Pavlos |
| author_facet | Konstantos, Konstantinos Motakis, Pavlos |
| contents | For every $1\leq α<ω_1$, we construct an explicit unconditional finite-dimensional decomposition (FDD) $(X_λ)_{λ\in\mathcal{T}_α}$ of the Bourgain-Rosenthal-Schechtman space $R_α^{p,0}$ by blocking its standard martingale difference sequence (MDS) basis. This FDD has strong reproducing properties and supports a theory of distributional representations between the spaces $R_α^{p,0}$, $1\leq α<ω_1$. We use this framework to prove an approximate orthogonal reduction: every bounded linear operator on a limit space $R_α^{p,0}$ is, via a distributional embedding and up to arbitrary precision, reduced to a scalar FDD-diagonal operator. As a consequence, the standard MDS bases of the limit spaces $R_α^{p,0}$ satisfy the factorization property. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_24487 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Coordinate systems and distributional embeddings in Bourgain-Rosenthal-Schechtman spaces: a framework for operator reduction Konstantos, Konstantinos Motakis, Pavlos Functional Analysis 46B09, 46B25, 46B28, 46E30, 47A68 For every $1\leq α<ω_1$, we construct an explicit unconditional finite-dimensional decomposition (FDD) $(X_λ)_{λ\in\mathcal{T}_α}$ of the Bourgain-Rosenthal-Schechtman space $R_α^{p,0}$ by blocking its standard martingale difference sequence (MDS) basis. This FDD has strong reproducing properties and supports a theory of distributional representations between the spaces $R_α^{p,0}$, $1\leq α<ω_1$. We use this framework to prove an approximate orthogonal reduction: every bounded linear operator on a limit space $R_α^{p,0}$ is, via a distributional embedding and up to arbitrary precision, reduced to a scalar FDD-diagonal operator. As a consequence, the standard MDS bases of the limit spaces $R_α^{p,0}$ satisfy the factorization property. |
| title | Coordinate systems and distributional embeddings in Bourgain-Rosenthal-Schechtman spaces: a framework for operator reduction |
| topic | Functional Analysis 46B09, 46B25, 46B28, 46E30, 47A68 |
| url | https://arxiv.org/abs/2510.24487 |